Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries

Damien Allonsius,Franck Boyer

doi:10.3934/mcrf.2019037

Abstract

The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than \begin{document}$ 1 $\end{document} . We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general. The main assumption we make on the geometry and on the evolution equation itself is that it can be put into a tensorized form. In such a case, following [ 5 ] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisfies some necessary Kalman condition) with uniform bounds on the controls. To achieve this objective we actually propose an abstract result on ordinary differential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters.

Highlights

We are interested in null-controllability properties at any time T > 0 of coupled linear parabolic equations at the continuous level as well as at the semi-discrete in space level
Β = 0, on (0, T ) × ∂Ω, where Ω is a bounded domain of Rd (d ≥ 1), Γ is a non empty part of the boundary ∂Ω, α and β are the two components of the system, and v is the boundary control we are looking for
The main difficulty in the analysis of the controllability of such system comes from the fact that we only have one boundary control v to drive the two components (α, β) to 0 at the final time

Summary

Introduction

It is well-known (see [10, 15] for instance) that System (11) is null controllable at time T for any initial condition y0 if and only if there exists C > 0, such that the following observability inequality for the adjoint problem is satisfied e−T L∗ qT Let Di be two positive definite self-adjoint operators in those spaces and for any s ∈ R, we introduce the following scalar products ui, vi s,Di = Dsi ui, vi 0,i , ∀ui, vi ∈ Ei, and the associated norms

Objectives

Results

Conclusion